Surjectivity of mod 2^n representations of elliptic curves

Abstract

For an elliptic curve E over Q, the Galois action on the l-power torsion points defines representations whose images are subgroups of GL_2(Z/l^n Z). There are three exceptional prime powers l^n=2,3,4 when surjectivity of the mod l^n representation does not imply that for l^(n+1). Elliptic curves with surjective mod 3 but not mod 9 representation have been classified by Elkies. The purpose of this note is to do this in the other two cases.

3 pages

References in corpus (1)

• Elliptic curves with 3-adic Galois representation surjective mod 3 but not mod 9

Cited by in corpus (17)

• Elliptic Curves with abelian division fields
• A classification of isogeny-torsion graphs of $\mathbb{Q}$-isogeny classes of elliptic curves
• Galois representations attached to elliptic curves with complex multiplication
• Torsion of rational elliptic curves over cubic fields
• Serre's constant of elliptic curves over the rationals
• On the surjectivity of mod $\ell$ representations associated to elliptic curves
• Elliptic curves with non-abelian entanglements
• A bound for the conductor of an open subgroup of GL2 associated to an elliptic curve
• Pathologies of the Brauer-Manin obstruction
• Lifting Subgroups of Symplectic Groups over $\mathbb{Z}/\ell \mathbb{Z}$
• On the class numbers of the fields of the $p^n$-torsion points of elliptic curves over $\mathbb{Q}$
• Elliptic curves with 2-torsion contained in the 3-torsion field
• Arithmetic of quaternion origami
• Near coincidences and nilpotent division fields
• Elliptic curves with missing Frobenius traces
• Elliptic curves over $\mathbb{Q}$ and 2-adic images of Galois
• The density of odd order reductions for elliptic curves with a rational point of order 2